
Based on the function space of identifiability, we introduce next a nonparametric learning algorithm with SIDA-RKHS regularization. The algorithm consists of three steps.
First, we utilize the data to estimate the exploration measure and the support of the kernel. Based on them, we set a class of hypothesis spaces, with their dimensions, i.e., the number of basis functions, in a proper range moving from under-fitting to over-fitting. For the hypothesis space \(\mathcal {H}_n =\mathrm {span}\lbrace \phi _i\rbrace _{i=1}^n\) , we compute the basis matrix \(B_n = ( \langle {\phi _i,\phi _j}\rangle _{L^2(\rho _N^J)})_{1\le i,j\le n} \in \mathbb {R}^{n\times n}\) .
Second, we assemble the regression matrices from data for each of these hypothesis spaces. We approximate the integrals by Riemann sum or other numerical integrator.
Finally, we identify an estimator with SIDA-RKHS regularization for each of these hypothesis spaces by the L-curve method [1]} and select the one with the best fitting.
We summarize the method in Algorithm REF , with its full details provided in Section .
